This project develops and studies statistical confidence bounds useful for low-dose extrapolation in quantitative risk analysis. Application is intended for those risk assessment studies where human or animal data are used to set benchmark or other safe low-dose levels of a toxic stimulus, but where study information is limited to high dose levels of the stimulus. Methods are derived for estimating upper confidence limits on predicted risk and on predicted additional risk for various endpoints, measured on both continuous and discrete scales. From the simultaneous confidence bounds, lower confidence limits on the "effective" stimulus or dose (ED) associated with a particular risk are calculated. An important feature of the simultaneous construction is that any inferences based on inverting the simultaneous confidence bounds apply automatically to inverse bounds on the ED. The methodology extends existing theory on simultaneous prediction bands which applies to mathematical prediction equations useful in quantitative risk assessment/low-dose extrapolation problems. New concepts developed in order to achieve these goals include simultaneous band optimization in low-dose regions of the dose scale, and application to non-normally distributed data and to non-linear, possibly non-monotone dose-response functions. Extensions to simultaneous bounds useful in unlimited inverse prediction for low-dose extrapolation also are considered. An evaluation phase of the project studies the small-sample operating characteristics of the new methodology via Monte Carlo computer calculations, and applies the new methods to existing data for a number of low-dose risk endpoints, including concentration levels of detrimental toxins or toxic metabolites in human or animal subjects, body weight losses in laboratory animals, mutation frequencies in transgenic animals and other transgenic systems, mutational spectra in human or animal/transgenic systems, carcinogenicity rates in laboratory animals, or reproductive capacity limitations in aquatic or other eco-toxicological systems. The new methods fill existing gaps in low-dose risk extrapolation, and have application to a wide variety of data-analytic scenarios in quantitative risk assessment.